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G = Q16.4D4order 128 = 27

1st non-split extension by Q16 of D4 acting via D4/C4=C2

p-group, metabelian, nilpotent (class 4), monomial

Aliases: C42Q32, Q16.4D4, C42.139D4, C4⋊C16.9C2, (C2×C8).68D4, C8.73(C2×D4), C2.5(C2×Q32), (C2×C4).150D8, (C2×Q32).2C2, (C4×Q16).5C2, C8.91(C4○D4), (C4×C8).64C22, (C2×C16).5C22, C4⋊Q16.8C2, C4.52(C4⋊D4), C2.21(C4⋊D8), C4.17(C8⋊C22), (C2×C8).517C23, C2.Q32.2C2, C22.103(C2×D8), (C2×Q16).3C22, C2.10(Q32⋊C2), C2.D8.159C22, (C2×C4).785(C2×D4), SmallGroup(128,941)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C8 — Q16.4D4
C1C2C4C8C2×C8C2×Q16C4×Q16 — Q16.4D4
C1C2C4C2×C8 — Q16.4D4
C1C22C42C4×C8 — Q16.4D4
C1C2C2C2C2C4C4C2×C8 — Q16.4D4

Generators and relations for Q16.4D4
 G = < a,b,c,d | a8=c4=1, b2=d2=a4, bab-1=dad-1=a-1, ac=ca, bc=cb, dbd-1=a-1b, dcd-1=c-1 >

Subgroups: 172 in 77 conjugacy classes, 34 normal (22 characteristic)
C1, C2 [×3], C4 [×2], C4 [×2], C4 [×6], C22, C8 [×2], C8, C2×C4 [×3], C2×C4 [×4], Q8 [×7], C16 [×2], C42, C42, C4⋊C4 [×4], C2×C8 [×2], Q16 [×2], Q16 [×7], C2×Q8 [×3], C4×C8, Q8⋊C4, C2.D8, C2×C16 [×2], Q32 [×4], C4×Q8, C4⋊Q8, C2×Q16, C2×Q16 [×2], C2×Q16, C2.Q32 [×2], C4⋊C16, C4×Q16, C4⋊Q16, C2×Q32 [×2], Q16.4D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D8 [×2], C2×D4 [×2], C4○D4, Q32 [×2], C4⋊D4, C2×D8, C8⋊C22, C4⋊D8, C2×Q32, Q32⋊C2, Q16.4D4

Character table of Q16.4D4

 class 12A2B2C4A4B4C4D4E4F4G4H4I4J4K8A8B8C8D8E8F16A16B16C16D16E16F16G16H
 size 1111222248888161622224444444444
ρ111111111111111111111111111111    trivial
ρ21111111111111-1-1111111-1-1-1-1-1-1-1-1    linear of order 2
ρ311111-1-11-1-1-111-111111-1-1-1111-1-1-11    linear of order 2
ρ411111-1-11-1-1-1111-11111-1-11-1-1-1111-1    linear of order 2
ρ511111-1-11-111-1-1-111111-1-11-1-1-1111-1    linear of order 2
ρ611111-1-11-111-1-11-11111-1-1-1111-1-1-11    linear of order 2
ρ7111111111-1-1-1-111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ8111111111-1-1-1-1-1-111111111111111    linear of order 2
ρ92-2-22200-202-200002-22-20000000000    orthogonal lifted from D4
ρ102-2-22200-20-2200002-22-20000000000    orthogonal lifted from D4
ρ11222222222000000-2-2-2-2-2-200000000    orthogonal lifted from D4
ρ1222222-2-22-2000000-2-2-2-22200000000    orthogonal lifted from D4
ρ132222-2-2-2-220000000000002-22-2-22-22    orthogonal lifted from D8
ρ142222-222-2-200000000000022-22-22-2-2    orthogonal lifted from D8
ρ152222-222-2-2000000000000-2-22-22-222    orthogonal lifted from D8
ρ162222-2-2-2-22000000000000-22-222-22-2    orthogonal lifted from D8
ρ1722-2-20-2200000000-2-222-22ζ165163ζ16516316716165163ζ1671616516316716ζ16716    symplectic lifted from Q32, Schur index 2
ρ1822-2-20-2200000000-2-222-22165163165163ζ16716ζ16516316716ζ165163ζ1671616716    symplectic lifted from Q32, Schur index 2
ρ1922-2-202-20000000022-2-2-2216716ζ16716ζ16516316716ζ165163ζ16716165163165163    symplectic lifted from Q32, Schur index 2
ρ2022-2-20-220000000022-2-22-21671616716165163ζ16716ζ165163ζ16716165163ζ165163    symplectic lifted from Q32, Schur index 2
ρ2122-2-202-200000000-2-2222-2165163ζ1651631671616516316716ζ165163ζ16716ζ16716    symplectic lifted from Q32, Schur index 2
ρ2222-2-202-200000000-2-2222-2ζ165163165163ζ16716ζ165163ζ167161651631671616716    symplectic lifted from Q32, Schur index 2
ρ2322-2-20-220000000022-2-22-2ζ16716ζ16716ζ1651631671616516316716ζ165163165163    symplectic lifted from Q32, Schur index 2
ρ2422-2-202-20000000022-2-2-22ζ1671616716165163ζ1671616516316716ζ165163ζ165163    symplectic lifted from Q32, Schur index 2
ρ252-2-22200-2000-2i2i00-22-220000000000    complex lifted from C4○D4
ρ262-2-22200-20002i-2i00-22-220000000000    complex lifted from C4○D4
ρ274-4-44-4004000000000000000000000    orthogonal lifted from C8⋊C22
ρ284-44-40000000000022-22-22220000000000    symplectic lifted from Q32⋊C2, Schur index 2
ρ294-44-400000000000-222222-220000000000    symplectic lifted from Q32⋊C2, Schur index 2

Smallest permutation representation of Q16.4D4
Regular action on 128 points
Generators in S128
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32)(33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64)(65 66 67 68 69 70 71 72)(73 74 75 76 77 78 79 80)(81 82 83 84 85 86 87 88)(89 90 91 92 93 94 95 96)(97 98 99 100 101 102 103 104)(105 106 107 108 109 110 111 112)(113 114 115 116 117 118 119 120)(121 122 123 124 125 126 127 128)
(1 25 5 29)(2 32 6 28)(3 31 7 27)(4 30 8 26)(9 19 13 23)(10 18 14 22)(11 17 15 21)(12 24 16 20)(33 59 37 63)(34 58 38 62)(35 57 39 61)(36 64 40 60)(41 51 45 55)(42 50 46 54)(43 49 47 53)(44 56 48 52)(65 90 69 94)(66 89 70 93)(67 96 71 92)(68 95 72 91)(73 82 77 86)(74 81 78 85)(75 88 79 84)(76 87 80 83)(97 122 101 126)(98 121 102 125)(99 128 103 124)(100 127 104 123)(105 114 109 118)(106 113 110 117)(107 120 111 116)(108 119 112 115)
(1 43 11 35)(2 44 12 36)(3 45 13 37)(4 46 14 38)(5 47 15 39)(6 48 16 40)(7 41 9 33)(8 42 10 34)(17 57 25 49)(18 58 26 50)(19 59 27 51)(20 60 28 52)(21 61 29 53)(22 62 30 54)(23 63 31 55)(24 64 32 56)(65 97 73 105)(66 98 74 106)(67 99 75 107)(68 100 76 108)(69 101 77 109)(70 102 78 110)(71 103 79 111)(72 104 80 112)(81 113 89 121)(82 114 90 122)(83 115 91 123)(84 116 92 124)(85 117 93 125)(86 118 94 126)(87 119 95 127)(88 120 96 128)
(1 105 5 109)(2 112 6 108)(3 111 7 107)(4 110 8 106)(9 99 13 103)(10 98 14 102)(11 97 15 101)(12 104 16 100)(17 123 21 127)(18 122 22 126)(19 121 23 125)(20 128 24 124)(25 115 29 119)(26 114 30 118)(27 113 31 117)(28 120 32 116)(33 67 37 71)(34 66 38 70)(35 65 39 69)(36 72 40 68)(41 75 45 79)(42 74 46 78)(43 73 47 77)(44 80 48 76)(49 83 53 87)(50 82 54 86)(51 81 55 85)(52 88 56 84)(57 91 61 95)(58 90 62 94)(59 89 63 93)(60 96 64 92)

G:=sub<Sym(128)| (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88)(89,90,91,92,93,94,95,96)(97,98,99,100,101,102,103,104)(105,106,107,108,109,110,111,112)(113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128), (1,25,5,29)(2,32,6,28)(3,31,7,27)(4,30,8,26)(9,19,13,23)(10,18,14,22)(11,17,15,21)(12,24,16,20)(33,59,37,63)(34,58,38,62)(35,57,39,61)(36,64,40,60)(41,51,45,55)(42,50,46,54)(43,49,47,53)(44,56,48,52)(65,90,69,94)(66,89,70,93)(67,96,71,92)(68,95,72,91)(73,82,77,86)(74,81,78,85)(75,88,79,84)(76,87,80,83)(97,122,101,126)(98,121,102,125)(99,128,103,124)(100,127,104,123)(105,114,109,118)(106,113,110,117)(107,120,111,116)(108,119,112,115), (1,43,11,35)(2,44,12,36)(3,45,13,37)(4,46,14,38)(5,47,15,39)(6,48,16,40)(7,41,9,33)(8,42,10,34)(17,57,25,49)(18,58,26,50)(19,59,27,51)(20,60,28,52)(21,61,29,53)(22,62,30,54)(23,63,31,55)(24,64,32,56)(65,97,73,105)(66,98,74,106)(67,99,75,107)(68,100,76,108)(69,101,77,109)(70,102,78,110)(71,103,79,111)(72,104,80,112)(81,113,89,121)(82,114,90,122)(83,115,91,123)(84,116,92,124)(85,117,93,125)(86,118,94,126)(87,119,95,127)(88,120,96,128), (1,105,5,109)(2,112,6,108)(3,111,7,107)(4,110,8,106)(9,99,13,103)(10,98,14,102)(11,97,15,101)(12,104,16,100)(17,123,21,127)(18,122,22,126)(19,121,23,125)(20,128,24,124)(25,115,29,119)(26,114,30,118)(27,113,31,117)(28,120,32,116)(33,67,37,71)(34,66,38,70)(35,65,39,69)(36,72,40,68)(41,75,45,79)(42,74,46,78)(43,73,47,77)(44,80,48,76)(49,83,53,87)(50,82,54,86)(51,81,55,85)(52,88,56,84)(57,91,61,95)(58,90,62,94)(59,89,63,93)(60,96,64,92)>;

G:=Group( (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88)(89,90,91,92,93,94,95,96)(97,98,99,100,101,102,103,104)(105,106,107,108,109,110,111,112)(113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128), (1,25,5,29)(2,32,6,28)(3,31,7,27)(4,30,8,26)(9,19,13,23)(10,18,14,22)(11,17,15,21)(12,24,16,20)(33,59,37,63)(34,58,38,62)(35,57,39,61)(36,64,40,60)(41,51,45,55)(42,50,46,54)(43,49,47,53)(44,56,48,52)(65,90,69,94)(66,89,70,93)(67,96,71,92)(68,95,72,91)(73,82,77,86)(74,81,78,85)(75,88,79,84)(76,87,80,83)(97,122,101,126)(98,121,102,125)(99,128,103,124)(100,127,104,123)(105,114,109,118)(106,113,110,117)(107,120,111,116)(108,119,112,115), (1,43,11,35)(2,44,12,36)(3,45,13,37)(4,46,14,38)(5,47,15,39)(6,48,16,40)(7,41,9,33)(8,42,10,34)(17,57,25,49)(18,58,26,50)(19,59,27,51)(20,60,28,52)(21,61,29,53)(22,62,30,54)(23,63,31,55)(24,64,32,56)(65,97,73,105)(66,98,74,106)(67,99,75,107)(68,100,76,108)(69,101,77,109)(70,102,78,110)(71,103,79,111)(72,104,80,112)(81,113,89,121)(82,114,90,122)(83,115,91,123)(84,116,92,124)(85,117,93,125)(86,118,94,126)(87,119,95,127)(88,120,96,128), (1,105,5,109)(2,112,6,108)(3,111,7,107)(4,110,8,106)(9,99,13,103)(10,98,14,102)(11,97,15,101)(12,104,16,100)(17,123,21,127)(18,122,22,126)(19,121,23,125)(20,128,24,124)(25,115,29,119)(26,114,30,118)(27,113,31,117)(28,120,32,116)(33,67,37,71)(34,66,38,70)(35,65,39,69)(36,72,40,68)(41,75,45,79)(42,74,46,78)(43,73,47,77)(44,80,48,76)(49,83,53,87)(50,82,54,86)(51,81,55,85)(52,88,56,84)(57,91,61,95)(58,90,62,94)(59,89,63,93)(60,96,64,92) );

G=PermutationGroup([(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32),(33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64),(65,66,67,68,69,70,71,72),(73,74,75,76,77,78,79,80),(81,82,83,84,85,86,87,88),(89,90,91,92,93,94,95,96),(97,98,99,100,101,102,103,104),(105,106,107,108,109,110,111,112),(113,114,115,116,117,118,119,120),(121,122,123,124,125,126,127,128)], [(1,25,5,29),(2,32,6,28),(3,31,7,27),(4,30,8,26),(9,19,13,23),(10,18,14,22),(11,17,15,21),(12,24,16,20),(33,59,37,63),(34,58,38,62),(35,57,39,61),(36,64,40,60),(41,51,45,55),(42,50,46,54),(43,49,47,53),(44,56,48,52),(65,90,69,94),(66,89,70,93),(67,96,71,92),(68,95,72,91),(73,82,77,86),(74,81,78,85),(75,88,79,84),(76,87,80,83),(97,122,101,126),(98,121,102,125),(99,128,103,124),(100,127,104,123),(105,114,109,118),(106,113,110,117),(107,120,111,116),(108,119,112,115)], [(1,43,11,35),(2,44,12,36),(3,45,13,37),(4,46,14,38),(5,47,15,39),(6,48,16,40),(7,41,9,33),(8,42,10,34),(17,57,25,49),(18,58,26,50),(19,59,27,51),(20,60,28,52),(21,61,29,53),(22,62,30,54),(23,63,31,55),(24,64,32,56),(65,97,73,105),(66,98,74,106),(67,99,75,107),(68,100,76,108),(69,101,77,109),(70,102,78,110),(71,103,79,111),(72,104,80,112),(81,113,89,121),(82,114,90,122),(83,115,91,123),(84,116,92,124),(85,117,93,125),(86,118,94,126),(87,119,95,127),(88,120,96,128)], [(1,105,5,109),(2,112,6,108),(3,111,7,107),(4,110,8,106),(9,99,13,103),(10,98,14,102),(11,97,15,101),(12,104,16,100),(17,123,21,127),(18,122,22,126),(19,121,23,125),(20,128,24,124),(25,115,29,119),(26,114,30,118),(27,113,31,117),(28,120,32,116),(33,67,37,71),(34,66,38,70),(35,65,39,69),(36,72,40,68),(41,75,45,79),(42,74,46,78),(43,73,47,77),(44,80,48,76),(49,83,53,87),(50,82,54,86),(51,81,55,85),(52,88,56,84),(57,91,61,95),(58,90,62,94),(59,89,63,93),(60,96,64,92)])

Matrix representation of Q16.4D4 in GL4(𝔽17) generated by

14300
141400
00160
00016
,
71600
161000
001115
0096
,
1000
0100
00109
0027
,
0400
4000
00109
0067
G:=sub<GL(4,GF(17))| [14,14,0,0,3,14,0,0,0,0,16,0,0,0,0,16],[7,16,0,0,16,10,0,0,0,0,11,9,0,0,15,6],[1,0,0,0,0,1,0,0,0,0,10,2,0,0,9,7],[0,4,0,0,4,0,0,0,0,0,10,6,0,0,9,7] >;

Q16.4D4 in GAP, Magma, Sage, TeX

Q_{16}._4D_4
% in TeX

G:=Group("Q16.4D4");
// GroupNames label

G:=SmallGroup(128,941);
// by ID

G=gap.SmallGroup(128,941);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,448,141,64,422,352,1684,438,242,4037,1027,124]);
// Polycyclic

G:=Group<a,b,c,d|a^8=c^4=1,b^2=d^2=a^4,b*a*b^-1=d*a*d^-1=a^-1,a*c=c*a,b*c=c*b,d*b*d^-1=a^-1*b,d*c*d^-1=c^-1>;
// generators/relations

Export

Character table of Q16.4D4 in TeX

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